Question: Let $S$ be a surface in 3D described by the equation $x^3 - yz + \sin(x) = \pi^3 - 20$. What is the equation of the plane tangent to $S$ at $(\pi, 4, 5)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(3\pi^2 - 1) (x - \pi) - 5(y - 4) - 4(z - 5) = 0$ (Choice B) B $(3\pi^2 - 1) (x + \pi) - 5(y + 4) - 4(z + 5) = 0$ (Choice C) C $\pi (x - 3\pi^2 + 1) + 4(y + 5) + 5(z + 4) = 0$ (Choice D) D $3 \pi^2 (x - \pi) + 4(y - 5) + 5(z - 4) = 0$
Answer: The equation for a tangent plane of an implicitly defined surface $F(x, y, z) = 0$ at the point $(a, b, c)$ is: $F_x(x - a) + F_y(y - b) + F_z(z - c) = 0$ [What's the intuition behind the formula?] Let's find $F_x$, $F_y$, and $F_z$. $\begin{aligned} F_x &= 3x^2 + \cos(x) = 3\pi^2 - 1\\ \\ F_y &= -z = -5 \\ \\ F_z &= -y = -4 \end{aligned}$ Putting it all together, here's the equation for the tangent plane of $S$ at $(\pi, 4, 5)$ : $(3\pi^2 - 1) (x - \pi) - 5(y - 4) - 4(z - 5) = 0$